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# How To Get The Eight Answer

Author: N.H. Crowhurst

We like to use a lot of feedback but this results in conditions, if we are not careful, that cause oscillation, because A|3 will be so big at the starting point that it will be difficult to have it turn in sharply enough to get "inside" the 1 point. We can look at this on the graphs of db response and phase shift caused by each coupling network. We notice that, at the 3-db point, the phase shift is just 45°. The phase shift never quite reaches 90° but the db response keeps on falling off at almost 6 db for every octave in frequency.

If we add a second arrangement with the 3-db point at the same frequency, we would double these values. There would be a 6-db loss of magnitude and a 90° phase shift. The phase shift would never quite reach 180° but the db response keeps on falling off at almost 12 db for every octave. Although this two-stage arrangement can never get as far as causing oscillation, it can cause a peak in the frequency response.

This can be shown easily by drawing this curve on top of the background of circles representing constant ratios. Changing the size of the two-stage curve alters it from a condition where it follows one of the circles and then turns in to the "back" of the point O to one where it moves outward over the circles before cutting back across them. When it gets as big as this, the response indicated is one with a peak.

 Curves for two stages

If more than two stages are used, the ultimate phase shift approaches 270°, so it must pass through 180° somewhere. The problem now is to make sure that the amplification has reduced, so A(5 is less than 1, by the time the phase shift reaches 180°.

 The high frequency end of the response curves

The best way of achieving this proves to be a choice in the combination of coupling capacitors values (for the low-frequency response) that causes one rolloff or 3-db point to occur at a much higher frequency than the other two (or more).

 The low frequency end of the response curves

In this way the gain is reduced at the rate of 6 db for every octave with only a little more than 90° phase shift (because the other rolloffs are gradually starting) so it gets the magnitude of A[J down to much less than 1 before the phase shift reaches 180°.

If similar rolloff points are used at each stage, a three-stage amplifier will start to oscillate when A(J reaches 18 db (or a loop gain of 8); four stages will only reach a loop gain of 4 (12 db); and five stages limit it to 3 (9.5 db).

Last Update: 2010-11-03